9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-4_661_915_932_431}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians.
Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),
- show that the perimeter, \(P\) cm, of the design is given by
$$P = 2 x + \frac { 100 } { x }$$
- Find the value of \(x\) for which \(P\) is a minimum.
- Show that \(P\) is a minimum for this value of \(x\).
- Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).